Android 6.0 needs additional code in AirDroid to grant access to the external "SD CARD", so the user has to confirm the access at least once for the application. There is no known way to get it working, until devs are doing the described, additional code for it. The way accessing the external "SD CARD" simply went to a complete other direction at Marshmallow (Android 6.0).

@Riccardo: With "SD CARD", you won't get into your external one. You need to have the button for it separately. So until the devs don't fix it for Android 6.0, it won't work. Simple.